Stress Isotropization in Weakly Jammed Granular Packings

Abstract

When sheared, granular media experience localized plastic events known as shear transformations which generate anisotropic internal stresses. Under strong confining pressure, the response of granular media to local force multipoles is essentially linear, resulting in quadrupolar propagated stresses. This can lead to additional plastic events along the direction of relative stress increase. Closer to the unjamming transition however, as the confining pressure and the shear modulus vanish, nonlinearities become relevant. Yet, the consequences of these nonlinearities on the stress response to plastic events remains poorly understood. We show with granular dynamics simulations that this brings about an isotropization of the propagated stresses, in agreement with a previously developed continuum elastic model. This could significantly modify the yielding transition of weakly-jammed amorphous media, which has been conceptualized as an avalanche of such plastic events.

Figures (5)

• Figure 1: The far-field stresses induced by a shear transformation (ST) become more isotropic near unjamming. (a) In an amorphous medium, a shear transformation, e.g., a local change of neighbors between grains (inset) Kabla03Desmond15, applies a local force dipole (orange arrowheads) on the surrounding medium. These forces propagate through the medium (grey arrows), resulting in stresses at the medium's boundary (blue arrowheads). (b) Far from unjamming, the medium propagates stresses according to linear elasticity. The symmetry of this stress response is thus independent of the magnitude of the local forces. (c) Close to unjamming, the medium may not support the propagation of tensile stresses, resulting in a dilational stress response. (d) For large local forces, stress redistribution within the medium results in an increasingly isotropic dilation.
• Figure 2: We subject circular jammed packings to small internal forces. (a) Packing of the type used in our simulations but with fewer disks. We exert radial forces on the disks in the shaded region near $r_\text{in}$ (orange), and measure the forces exerted on the disks in the shaded region near $r_\text{out}$ (blue). (b) The same packing under isotropic contractile forcing, $\mathcal{P}_l<0$, corresponding to a local shrinkage of the original orange ring. In the final configuration, some gray disks are now subject to the forcing, and some orange ones are not. The dashed circle has radius $r_\text{in}$. (c) Dipolar forcing, $\mathcal{S}_l>0$.
• Figure 3: Force chains rearrange to create boundary dilation out of local shear stress. (a) Local shear stress rearranges force chains in a packing. The green segments have widths and colors proportional to the forces between neighboring disks. We refer to consecutive segments with large widths as force chains. Here, we have $\simeq 6700$ disks, $r_\text{out}\simeq 44$, $r_\text{out}/r_\text{in}=3$, $\Delta\phi\simeq0.03$. (b) Dilation under dipolar forcing for three initial configurations (circles, squares and triangles), demonstrating reproducibility. We fit the data using $\mathcal{P}_b=\alpha\mathcal{S}_l^2$ with $\alpha\simeq 4400$, and $\mathcal{S}_b=(1+B)\mathcal{S}_l$ with $B\simeq0.7$. Here, $r_\text{out}/r_\text{in}=8$, $r_\text{in}\simeq 5.5$, $\Delta\phi\simeq 0.03$. (c) The anisotropy in the stress response decreases as the local shear stress increases. The vertical line indicates the local shear stress at which this anisotropy falls under 1/2.
• Figure 4: Isotropization prevails close to unjamming and in large systems. (a) Log-log plot of coefficients $\alpha$ and $\beta$ obtained as in the fits of Fig. \ref{['fig:gran_dip']}(b) showing a $(\Delta\phi)^{-2}$ divergence as unjamming is approached. The black line shows our theoretical predictions [Eq. \ref{['eq:alpha']}] and we observe $\beta\simeq\alpha/9$. Inset: the phenomenological coefficient $B$ does not strongly depend on $\Delta\phi$. Here, $r_\text{in}\simeq22$ and $r_\text{out}\simeq44$. (b) Holding the outer radius $r_\text{out}\simeq44$ constant, the isotropization coefficient $\alpha$ is strongest for small $r_\text{in}$, in agreement with the theoretical prediction (End Matter). Inset:$B$ is also large for small $r_\text{in}$; the dashed line shows a heuristic dependence $B=(r_\text{out}/r_\text{in}-1)/8$. Here, $\Delta\phi\simeq0.03$. Bars show standard deviation across three simulations.
• Figure 5: Theoretical prediction for the isotropization coefficient. Equation \ref{['eq:alpha']} predicts different scalings of $\alpha$ with $\Delta\phi$ for varying values of $r_\text{out}/r_\text{in}$. For area fractions $\Delta\phi\in[10^{-3},10^{-1}]$, the exponent varies from $-2$ when $r_\text{out}/r_\text{in}=10$, to $-1.5$ when $r_\text{out}/r_\text{in}=1.3$.